Hi Nemix,
I'm new to Unity myself, though it kind of controls like Maya so I didn't have much of a problem getting to grips with it. For my characters I tried various methods of exporting them from Maya as FBXs (baking the animations first etc) nothing seems to work right so I tried just copying my Maya .ma (MAYA ASCII) file into Unity and it works perfectly for me, I'm not longer losing body parts or any other errors.
I haven't tried Blender with this method but if Unity can recognise your constraints and such from .ma files I'm sure blender files shouldn't be a problem. Just make sure you bake all the animations to the skeleton and remove everything from the "character world"(and delete it) or whatever you named it but still have them grouped as they were. The reason for this is I found that Unity only searches one group down for components so if you have the characters group in the character world (treated as a group) your constraints wouldn't be recognized (or so i think, i might be very wrong here but it gave me problems til I deleted the world). Strangely enough it still recognized the geometry inside the group.
Not sure but I think Unity is a little sketchy when trying to read heavy FBXs so just try and import your model to it as is. The way I found out to use my Maya files was by making a poly cylinder, adding bones to it and doing some silly quick animations for idle, walking jumping and such to test different ways of exporting, might be worth it to do that before you start with your character so you know what works and save time.
If you aren't importing your model with animations and just wanted to bring it in to use mecanim I'm not too sure there but I think there are a few tutorials online for mecanim 
DISCLAIMER: I'm still a Unity noob so someone might have a better solution but for now this works for me and I'll stick to it
